Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x^{2}–3x+7
Ans: The equation 4x^{2}–3x+7 can be written as 4x^{2 }– 3x^{1 }+ 7x^{0}
Since x is the only variable in the given equation and the powers of x (i.e., 2, 1 and 0) are whole numbers, we can say that the expression 4x^{2 }– 3x + 7 is a polynomial in one variable.
(ii) y^{2}+√2
Ans: The equation y^{2 }+ √2 can be written as y^{2 }+ √2y^{0}
Since y is the only variable in the given equation and the powers of y (i.e., 2 and 0) are whole numbers, we can say that the expression y^{2 }+ √2 is a polynomial in one variable.
(iii) 3√t + t√2
Ans: The equation 3√t + t√2 can be written as 3t^{1/2 }+ √2t
Though t is the only variable in the given equation, the powers of t (i.e.,1/2) is not a whole number. Hence, we can say that the expression 3√t + t√2 is not a polynomial in one variable.
(iv) y + 2/y
Ans: The equation y + 2/y can be written as y + 2y^{1}
Though y is the only variable in the given equation, the powers of y (i.e.,1) is not a whole number. Hence, we can say that the expression y + 2/y is not a polynomial in one variable.
(v) x^{10 }+ y^{3 }+ t^{50}
Ans: Here, in the equation x^{10 }+ y^{3 }+ t^{50}
Though the powers, 10, 3, 50, are whole numbers, there are 3 variables used in the expression
x^{10 }+ y^{3 }+ t^{50}.
Hence, it is not a polynomial in one variable.
Q2. Write the coefficients of x^{2} in each of the following:
(i) 2 + x^{2 }+ x
Ans: The equation 2 + x^{2}+x can be written as 2 + (1)x^{2 }+ x
We know that, coefficient is the number which multiplies the variable.
Here, the number that multiplies the variable x^{2} is 1
the coefficients of x^{2} in 2 + x^{2 }+ x is 1.
(ii) 2 – x^{2 }+ x^{3}
Ans: The equation 2 – x^{2 }+ x^{3} can be written as 2 + (–1)x^{2 }+ x^{3}
We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.
Here, the number that multiplies the variable x^{2} is 1 the coefficients of x^{2} in 2 – x^{2 }+ x^{3} is 1.
(iii) (π/2)x^{2 }+ x
Ans: The equation (π/2)x^{2} + x can be written as (π/2)x^{2} + x
We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.
Here, the number that multiplies the variable x^{2} is π/2.
the coefficients of x^{2} in (π/2)x^{2} +x is π/2.
(iv)√2x  1
Ans: The equation √2x  1 can be written as 0x^{2}+√2x1 [Since 0x^{2} is 0]
We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.
Here, the number that multiplies the variable x^{2} is 0, the coefficients of x^{2} in √2x  1 is 0.
Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Ans: The degree of a polynomial is the highest power of the variable in the polynomial. It represents the highest exponent of the variable within the algebraic expression.
Therefore,
Q4. Write the degree of each of the following polynomials:
(i) 5x^{3 }+ 4x^{2 }+ 7x
Ans: The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 5x^{3 }+ 4x^{2 }+ 7x = 5x^{3 }+ 4x^{2 }+ 7x^{1}
The powers of the variable x are: 3, 2, 1
The degree of 5x^{3 }+ 4x^{2 }+ 7x is 3 as 3 is the highest power of x in the equation.
(ii) 4 – y^{2}
Ans: The highest power of the variable in a polynomial is the degree of the polynomial.
Here, in 4–y^{2},
The power of the variable y is 2
The degree of 4 – y^{2} is 2 as 2 is the highest power of y in the equation.
(iii) 5t – √7
Ans: The highest power of the variable in a polynomial is the degree of the polynomial.
Here, in 5t–√7,
The power of the variable t is 1.
The degree of 5t–√7 is 1 as 1 is the highest power of y in the equation.
(iv) 3
Ans: The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 3 = 3 × 1 = 3 × x^{0}
The power of the variable here is: 0
The degree of 3 is 0.
Q5. Classify the following as linear, quadratic and cubic polynomials:
Ans: We know that,
Linear polynomial: A polynomial of degree one is called a linear polynomial.
Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial.
Cubic polynomial: A polynomial of degree three is called a cubic polynomial.
(i) x^{2 }+ x
Ans: The highest power of x^{2 }+ x is 2
The degree is 2
Hence, x^{2 }+ x is a quadratic polynomial
(ii) x – x^{3}
Ans: The highest power of x–x^{3} is 3
The degree is 3
Hence, x–x^{3} is a cubic polynomial
(iii) y + y^{2 }+ 4
Ans: The highest power of y+y^{2}+4 is 2
The degree is 2
Hence, y+y^{2}+4is a quadratic polynomial
(iv) 1 + x
Ans: The highest power of 1 + x is 1
The degree is 1
Hence, 1 + x is a linear polynomial.
(v) 3t
Ans: The highest power of 3t is 1
The degree is 1
Hence, 3t is a linear polynomial.
(vi) r^{2}
Ans: The highest power of r^{2} is 2
The degree is 2
Hence, r^{2 }is a quadratic polynomial.
(vii) 7x^{3}
Ans: The highest power of 7x^{3} is 3
The degree is 3
Hence, 7x^{3} is a cubic polynomial.
1. What are polynomials? 
2. What is the degree of a polynomial? 
3. What is the remainder theorem of polynomials? 
4. How do we find the roots of a polynomial equation? 
5. What is the factor theorem of polynomials? 
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62 videos426 docs102 tests
